Completely Controlling the Dimensions of Formal Fiber Rings at Prime Ideals of Small Height

TL;DR

This paper constructs local UFDs with prescribed formal fiber dimensions at prime ideals, extending control over their algebraic properties, especially in the context of regular, excellent rings in characteristic zero.

Contribution

It introduces a method to explicitly control the dimensions of formal fibers at prime ideals in local UFDs, including the construction of excellent rings in characteristic zero.

Findings

Constructed local UFDs with prescribed formal fiber dimensions

Achieved control over formal fibers at prime ideals of various heights

Extended results to regular and excellent rings in characteristic zero

Abstract

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.